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Sequences and Series question

2018 · 15 Apr · Shift 1 · Q33
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Sequences and Series question

2018 · 15 Apr · Shift 1 · Q33

JEE MainMathematicsSequences and SeriesMCQ+4 / −1
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :
  1. A
    (−∞-\infty−∞, −-− 10]
  2. B
    (−-− 10, 0)
  3. C
    (0, 10)
  4. D
    [10, ∞\infty∞)
View written solutionFree

Correct answer: C

  1. Let the first term of the infinite G.P. be bbb and common ratio be rrr.

  2. Since the sum of an infinite G.P. exists and is equal to 555, we must have ∣r∣<1|r|<1∣r∣<1 and the sum is S=b1−r=5.S=\frac{b}{1-r}=5.S=1−rb​=5.

  3. Therefore, b=5(1−r).b=5(1-r).b=5(1−r).

  4. Now use the condition ∣r∣<1|r|<1∣r∣<1, i.e. −1<r<1.-1<r<1.−1<r<1.

  5. Subtract throughout from 111: 0<1−r<2.0<1-r<2.0<1−r<2.

  6. Multiply by 555: 0<5(1−r)<10.0<5(1-r)<10.0<5(1−r)<10.

  7. But b=5(1−r)b=5(1-r)b=5(1−r), so 0<b<10.0<b<10.0<b<10.

  8. Hence bbb lies in the interval (0,10).(0,10).(0,10).

  9. Checking options:

    • A: (−∞,−10](-\infty,-10](−∞,−10] ❌
    • B: (−10,0)(-10,0)(−10,0) ❌
    • C: (0,10)(0,10)(0,10) ✅
    • D: [10,∞)[10,\infty)[10,∞) ❌

Therefore, the correct option is C.

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